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With some regularity conditions maximum likelihood estimators (MLEs) alwaysproduce asymptotically optimal (in the sense of consistency, efficiency,sufficiency, and unbiasedness) estimators. But in general, the MLEs lead tonon-robust statistical inference, for example, pricing models and riskmeasures. Actuarial claim severity is continuous, right-skewed, and frequentlyheavy-tailed. The data sets that such models are usually fitted to containoutliers that are difficult to identify and separate from genuine data.Moreover, due to commonly used actuarial "loss control strategies" in financialand insurance industries, the random variables we observe and wish to model areaffected by truncation (due to deductibles), censoring (due to policy limits),scaling (due to coinsurance proportions) and other transformations. Toalleviate the lack of robustness of MLE-based inference in risk modeling, herein this paper, we propose and develop a new method of estimation - method oftruncated moments (MTuM) and generalize it for different scenarios of losscontrol mechanism. Various asymptotic properties of those estimates areestablished by using central limit theory. New connections between differentestimators are found. A comparative study of newly-designed methods with thecorresponding MLEs is performed. Detail investigation has been done for asingle parameter Pareto loss model including a simulation study.

Truncated, Censored, and Actuarial Payment-type Moments for Robust Fitting of a Single-parameter Pareto Distribution